The control problems involved in paper-making processes can be divided into machine-directional (MD) control and cross-directional (CD) control. MD control concerns the paper properties along the machine direction, and a lot of control strategies have been reported and implemented.
CD control aims to reduce the variability of the paper property along the cross direction and to tune the dynamical property to meet the end users' specifications. The paper property is measured by a scanner mounted downstream traversing back and forth across the paper sheet; various feedback control strategies are proposed to achieve consistency of the paper profile. CD control is a challenging control problem that may involves hundreds of process actuators and hundreds or thousands of process measurements, and process models typically have a large amount of uncertainty associated with them. There are spatial and temporal aspects to this problem. The spatial aspect relates to variability of the process measurements across the sheet while the temporal aspect relates to variability of each process measurement over time.
Model predictive control (MPC), a control strategy which takes control and state constraints explicitly into consideration, has seen thousands of applications in industry, and has been recently introduced into CD control in paper-making processes with the advance of computational capability as well as the development of fast QP solvers. The spatially-distributed CD process is a two-dimensional (spatial and temporal) frequency process, and the spatial response and temporal response are decoupled. Consequently, the controller tuning of CD processes can be separated into spatial tuning and temporal tuning. Spatial tuning aims to tune the weighting matrices such that the steady-state paper property across the paper sheet is consistent; temporal tuning concerns more about the satisfaction of performance indices in terms of settling times and overshoots.
Many research results on spatial tuning have been reported. The spatial frequency concept of spatially-distributed CD processes has been investigated and the application of the spatial frequency has been proposed for the controller design of CD processes. It has been reported that the spatial frequency response of a single actuator determined the spatial frequency bandwidth of a CD process. A constructive procedure to design spatially-distributed feedback controllers has been applied in paper-making processes. Some stability margin and parameter tuning criteria were obtained via rectangular circulant matrices (RCMs) for the unconstrained CD-MPC, which provided a guide in the parameter tuning algorithms. Furthermore, an approximate steady-state performance prediction technique was proposed to speed up the parameter tuning procedure for the constrained CD-MPC. An automated tuning method was presented for the CD process such that the performance and robustness could be simultaneously satisfied under unstructured uncertainty.
In CD control design, the spatial and temporal aspects of the control problem are normally handled independently. For the spatial aspect of the problem robust control performance is desired. This means that the control will remain stable and will keep the variability of the CD profile small in spite of uncertainty about the process model. However, there is lack of an easy-to-use technique to tune a CD controller to provide a robust spatial control performance in the presence of parametric model uncertainty. Besides, as the CD system has a limited spatial bandwidth, it is also desirable that the limited spatial bandwidth property be explicitly incorporated into the parameter tuning procedure.